Axiom:Axiom of Replacement

Axiom

For any function $f$ and subset $S$ of the domain of $f$, there is a set containing the image $\map f S$.

More formally, let us express this as follows:

Let $\map P {x, z}$ be a propositional function, which determines a function.

That is, we have:

$\forall x: \exists ! y : \map P {x, y}$.

Then we state as an axiom:

$\forall A: \exists B: \forall y: \paren {y \in B \iff \exists x \in A : \map P {x,y}}$

Also presented as

The two above statements may be combined into a single (somewhat unwieldy) expression:

$\forall x: \exists ! y : \map P {x, y} \implies \forall A: \exists B: \forall y: \paren {y \in B \iff \exists x \in A : \map P {x,y}}$

Also known as

The axiom of replacement is also known as the axiom of substitution.

Historical Note

The axiom of replacement was added to the axioms of Zermelo set theory by Abraham Halevi Fraenkel, and also independently by Thoralf Albert Skolem.

The resulting system of axiomatic set theory is now referred to as Zermelo-Fraenkel Set Theory.

Sources

• 1982: Alan G. Hamilton: Numbers, Sets and Axioms ... (previous) ... (next): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF7}$
• 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Replacement Schema
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 9$ Zermelo set theory

• Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html